Recent Advances in Deep Learning-Based Reconstruction Pipeline Optimization for Magnetic Particle Imaging

Abstract

Magnetic particle imaging (MPI) is an emerging tracer-based imaging modality characterized by high sensitivity, quantitative linearity, and radiation-free operation. Despite these advantages, its performance is strongly influenced by many factors, including the system matrix (SM) calibration efficiency, reconstruction stability, noise suppression, and spatial resolution limitations. These technical challenges affect the reliability and robustness of MPI in biomedical imaging tasks such as tumor detection, vascular imaging, intracranial hemorrhage monitoring, and cell tracking. This review provides a structured overview of recent advances in MPI, with a particular focus on how deep learning techniques can be integrated into different stages of the reconstruction pipeline to address practical imaging challenges. Unlike existing reviews that mainly emphasize MPI principles, tracer development, or biomedical applications, this review organizes recent progress from the perspective of reconstruction pipeline optimization, covering signal acquisition and reconstruction foundations, X-space and SM-based methods, signal processing, SM calibration, inverse problem solving, and image post-processing. In addition, representative datasets and simulation platforms that support algorithm development and system evaluation are summarized. By connecting reconstruction theory and deep learning methodologies within a unified framework, this review highlights current methodological progress, key technical bottlenecks, and future opportunities for improving the stability, efficiency, and imaging performance of MPI.

Graphical abstract

Keywords

magnetic particle imaging / medical imaging / reconstruction / deep learning / biomedical applications

Introduction

Accurate visualization of biological processes plays a critical role in disease diagnosis, therapeutic monitoring, and biomedical research. Although conventional imaging modalities such as computed tomography (CT)[1,2], magnetic resonance imaging (MRI)[3,4], positron emission tomography (PET)[5,6], and fluorescence molecular imaging (FMI)[7,8] provide valuable anatomical and functional information, each modality faces inherent limitations related to radiation exposure, sensitivity, temporal resolution, or indirect contrast mechanisms. As imaging advances toward molecular and cellular scales[9], there is an increasing demand for techniques that enable highly sensitive, quantitative, and rapid visualization while minimizing potential side effects.

Magnetic particle imaging (MPI), first introduced by Gleich and Weizenecker in 2005[10], has emerged as a tracer-based imaging modality capable of directly detecting the nonlinear magnetic response of superparamagnetic iron oxide nanoparticles (SPIONs). By generating images without anatomical background signals and without ionizing radiation[11–14], MPI offers quantitative linearity and high temporal resolution. These properties have led to promising biomedical investigations, including cancer detection[15,16,17], intracranial hemorrhage detection[18], hyperthermia guidance[19], vascular imaging[20], and cell tracking[21].

Despite these advantages, the practical performance of MPI in biomedical applications is strongly dependent on engineering optimization. The reliability of tracer visualization, quantitative accuracy, and spatial resolution are governed by the reconstruction pipeline, which includes signal acquisition, signal processing, and image reconstruction. Current reconstruction strategies are broadly categorized into system matrix (SM)-based and X-space-based frameworks. While SM-based methods provide high-resolution imaging, they require extensive calibration measurements and are sensitive to noise due to the ill-conditioned nature of the inverse problem. X-space-based approaches offer computational efficiency but may face limitations in flexibility and resolution. These reconstruction-related constraints highlight that engineering innovation remains central to improving MPI performance.

In recent years, deep learning, a pivotal field in artificial intelligence[22], has significantly advanced domains such as computer vision[23,24] and natural language processing[25,26]. Recently, its techniques have increasingly been utilized as essential tools for medical or image data analysis[27,28]. In the medical domain, convolutional neural networks (CNNs)[30], recurrent neural networks (RNNs)[31], transformers[32], and algorithms such as semi-supervised[33] and weakly supervised learning[34] have been utilized. Additionally, the use of large-scale models also holds great promise for advancements in healthcare[28,29]. Deep learning techniques originating from medical tasks have also expanded into other domains, such as the U-Net[35] and ADMM-Net[36]. Moreover, deep learning has been applied to the field of medical imaging, including medical image reconstruction[37–39], signal processing[40], and imaging results analysis[41]. The recognized benefits of deep learning techniques can further improve MPI image quality and expand its biomedical applications. In this review, we summarize existing deep learning methods in the field of MPI, including MPI signal processing, SM super-resolution, SM-based image reconstruction, and image post-processing. Moreover, we systematically summarize the resources that have been made available to facilitate MPI research, including public datasets, simulation methods, and hardware system platforms. These resources are essential for advancing the field of MPI by providing researchers with readily accessible tools for experimentation, algorithm development, and system optimization. Finally, we discuss current progress and future directions for improving stability, efficiency, and overall imaging performance in MPI. To provide a clearer understanding of the overall structure of this review, a graphical overview is presented in Figure 1. This figure illustrates the reconstruction pipeline of MPI, including signal acquisition, SM-based and X-space-based reconstruction frameworks, and deep learning-assisted processing modules such as signal processing, SM super-resolution, image reconstruction, and image post-processing. It also serves as an organizational framework for the manuscript, helping to clarify how the subsequent sections are structured around different stages of the MPI reconstruction pipeline. The structure illustrated in Figure 1 corresponds to the organization of the following sections and provides a visual roadmap of the reconstruction workflow discussed throughout this review.

The remainder of this review is organized as follows. Section 2 introduces the fundamental principles of MPI signal acquisition and image reconstruction, including both X-space-based and SM-based reconstruction frameworks. Section 3 summarizes recent advances in deep learning-assisted MPI reconstruction, covering signal processing, SM super-resolution, SM-based reconstruction methods, image post-processing, and representative datasets and simulation resources that support algorithm development. Finally, Section 4 discusses current technical challenges and outlines future research directions for improving reconstruction stability, computational efficiency, and clinical applicability of MPI through engineering optimization and physics-informed learning strategies.

MPI Reconstruction

The MPI reconstruction process includes signal acquisition, signal processing, and image reconstruction. MPI generates signals through the interaction between magnetic nanoparticles (MNPs) and the external magnetic field, which contain information about the concentration and spatial location of MNPs. Since MPI signals contain a large amount of noise and redundant information, signal processing is carried out firstly in MPI reconstruction[42,43]. After processing the signals, an appropriate reconstruction algorithm is adopted according to the MPI device and noise level. Currently, the commonly used MPI reconstruction algorithms mainly include SM-based[43–45] and X-space-based reconstruction algorithms[46,47]. The following sections will introduce the generation and acquisition of MPI signals, signal processing, as well as image reconstruction algorithms based on the SM and X-space. Table 1 lists the adopted notation.

MPI signal acquisition

In an MPI system, the received current in the receiving coil is primarily induced by magnetic field changes. The key mathematical models describing the signal acquisition process are systematically summarized in Table 2. MPI generally employs a static magnetic field as the selection field, which does not generate electrical signals. Hence, the voltage signal u(t) in the receiving coil is mainly induced by the time-varying magnetization of particles and by the excitation field[42], as shown in Equation 1 in Table 2.

As stated by Faraday’s law of electromagnetic induction, the voltage induced in a single conductor is given by Equation 2 in Table 2.[48] Since B^H\left(r,t\right) is independent of the MNPs distribution, B^P\left(r,t\right) is primarily considered below. The magnetic flux density B^P\left(r,t\right) is expressible via the magnetic vector potential as \nabla \times A^P. Based on Stokes’ theorem, the voltage signal generated by magnetic particles is as shown in Equation 3 in Table 2,[40] with the signal generation process illustrated in Figure 2. Assuming the MNPs are placed far from the receiving coil, the magnetic vector potential can be expressed as the Equation 4 in Table 2.[42] Under ideal circumstances, it can be simplified by the Langevin function, as shown in Equation 5 in Table 2. According to Equation 4 and Equation 5, the voltage signal acquired by MPI can be obtained as the Equation 6.[42] The received signal is evidently influenced by the sample’s properties, the applied magnetic field, and the receiving coil.

This signal is captured by receive coils and, after filtering and amplification, results in the nonlinear response voltage signal. The induced voltage is the weighted sum of particle magnetizations from different positions, with the weights determined by the receive coil sensitivity, as shown in Equation 6 in Table 2. Moreover, the final detected voltage signal is the superposition of u^P and u^E. To extract u^P, there are two different methods, i.e., the compensation method and the filtering method, as shown in Figure 3. The compensation approach involves injecting a compensation signal into the receive chain to suppress u^E, as shown in Equation 7 in Table 2. A compensation coil, also known as a gradiometer, is employed to generate the same magnetic flux as the actual receive coil, as shown in Figure 3(A). This method was used in a spectrometric MPI scanner[49]. Nevertheless, the constraints of gradient compensation coils make their implementation in three-dimensional (3D) imaging difficult and ensuring that the compensation signal precisely matches the excitation signal across multiple orders of magnitude is a significant challenge. Consequently, the majority of MPI systems utilize an alternative filtering approach.

The filtering method is based on the principle that the particle signal and excitation signal occupy distinct frequency ranges in the frequency domain. In one-dimensional (1D) imaging, the spectrum of the excitation signal has only one frequency because it is a pure sinusoidal function, but the spectrum of the particle signal has many high-order harmonics. So, the detected signal is expanded into Fourier series, and the fundamental frequency is ignored, as shown in Equation 8. The spectrum comprises discrete lines at integer multiples of the frequency f^E, known as the fundamental frequency. f_k are frequency values of frequency components. In the hardware system, a band-stop filter as shown in Figure 3(B) is used to remove the excitation signal. Its function is to enhance signal quality and ensure that the excitation signal contains no high-order harmonics. For 3D imaging using Lissajous trajectory scanning, it is particularly convenient since the band-stop filter only requires tuning to the average excitation frequency, assuming the suppressed frequency bandwidth is adequately wide. However, it is important to acknowledge that this method may result in the loss of fundamental frequency signal.

MPI signal processing

MPI signal includes noise from both the environment and the device, which significantly affects the reconstructed images quality[50]. Therefore, during signal acquisition, multiple long-duration measurements followed by averaging are necessary[51]. While this method is highly effective in suppressing noise, it reduces temporal resolution, making it less suitable for biomedical applications like cardiovascular imaging[52], where high temporal resolution is critical. Moreover, to suppress the artifacts caused by background signal, background subtraction[53] is a commonly used method, in which background measurements are taken with an empty scanner during signal acquisition, and the background signal is then directly removed from the time-domain signal through subtraction. This method can only remove static background signals. For dynamically changing background signals, Straub et al.[54] proposed a joint reconstruction method for tracer distribution and background signal based on linear interpolation of background signals. This method is appropriate when the time interval between background and sample measurements is minimal, necessitating multiple background signal measurements and thereby reducing temporal resolution. A method for the combined estimation of background signals and particle distribution was introduced by Knopp et al.[44]. This approach derives a dictionary from many background scans using singular value decomposition and expresses the joint estimation as a linear Tikhonov regularization least squares problem. This method presumes a nearly linear drift in the background signal and requires a specific measurement protocol.

For SM-based reconstruction methods, after the signal is processed and converted to the frequency domain using Fourier transform, it is typically necessary to apply a predefined signal-to-noise ratio (SNR) threshold to filter the frequency components, using only those with high SNR for reconstruction[41]. Some researchers have also investigated frequency selection. Shan et al.[55] introduced a novel method termed SNR peak-based frequency selection (SPFS). In contrast to traditional SNR-based selection, SPFS focuses on frequencies with SNR peaks that capture essential information from the SM, leading to faster reconstruction times and enhanced spatial resolution. More recently, Zhu et al.[56] proposed an adaptive and robust frequency selection framework (AR-FSF) that automatically determines informative frequency components by incorporating velocity-corrected feature extraction and adaptive threshold estimation based on both SM and imaging signals, reducing dependence on empirically selected thresholds and improving reconstruction efficiency while maintaining high image quality across different imaging conditions.

The X-space method directly grids the time-domain signals into the reconstructed image[42]. Kurt et al.[57] addressed the issue of stitching together images from multiple separately reconstructed partial field of view (pFOV) images for the entire FOV. However, this process is sensitive to harmonic interference and noise. They introduced a robust X-space reconstruction technique named partial FOV center imaging (PCI) that efficiently mitigates harmonic interference. Table 3 summarizes the core formulas of MPI image reconstruction.

X-space-based reconstruction

The X-space reconstruction technique models the MPI system as an approximately linear shift-invariant (LSI) system characterized by a PSF[58]. This formulation enables rapid reconstruction by directly mapping time-domain signals to the spatial domain through gridding along the field-free point (FFP) trajectory. Under the LSI assumption, image pixel intensity maintains a direct correspondence with local tracer concentration. The original X-space method is derived based on several idealized assumptions, including the existence of a unique FFP, the adiabatic Langevin model, and the recovery of low-frequency signal components[59]. However, it should be noted that the LSI assumption represents an approximation in practical MPI systems. Factors such as particle relaxation effects, magnetic field inhomogeneities, trajectory distortions, and hardware imperfections may introduce spatially varying signal responses that deviate from strict shift invariance. These effects can lead to reconstruction inaccuracies such as spatial blurring, intensity distortions, and reduced quantitative reliability, particularly under non-ideal experimental conditions. The voltage signal expression for 1D MPI data acquired via Cartesian trajectory scanning is shown in Equation 1 in Table 3. The MPI image reconstruction task is to grid the induced voltage signal into the speed of the FFP, as shown in Equation 2 in Table 3.

In 2011, Goodwill and Conolly[60] expanded the 1D X-space theory to encompass two-dimensional (2D) and 3D X-space reconstruction. They demonstrated that 3D MPI adheres to an LSI imaging process and established its 3D PSF. The feasibility of the 3D X-space method was validated, though the reconstructed image quality still needed enhancement. In 2012, Goodwill et al.[46] introduced X-space MPI projection based on field free line (FFL) scanning theory. They employed permanent magnets to create the first FFL and produced the first FFL-based MPI image. To improve image quality, fast 3D MPI scanning was conducted using FFL-based projection X-space. In 2020, Kurtin[57] proposed the pFOV center imaging X-space reconstruction technique, which first generates a raw image of the full FOV by directly associating the MPI signal with the pFOV center. Then, deconvolution is applied to the raw image using a compact kernel to produce the final MPI image. Ozaslan et al.[61] introduced a universal reconstruction framework for X-space that is applicable to any scanning trajectory. This method automatically adjusts the reconstruction parameters without introducing additional blurring. They compared five different trajectories, demonstrating a significant improvement in image quality. Droigk et al.[62] introduced a frequency domain MPI reconstruction method based on a simplified Langevin paramagnetic model with certain approximations. This method employs Chebyshev polynomials of the second kind, offering fast reconstruction with image quality comparable to SM methods employing similar simplified physical assumptions.

The conventional X-space approach employs the Langevin model to characterize the magnetization of SPIONs under the adiabatic assumption. However, due to relaxation effects, this approach can introduce errors, resulting in inaccurate voltage signal simulation and image blurring. To address this issue, researchers have proposed more appropriate models to describe the magnetization process of SPIONs, aiming to improve image resolution. Croft et al.[63] presented the Debye model, adapting the X-space theory of MPI to incorporate nanoparticle relaxation effects. The Debye model approximates the magnetization as the time convolution of adiabatic magnetization with a kernel that represents the relaxation process, as shown in Equation 3 in Table 3. Nevertheless, solely relying on τ is inadequate for accurately characterizing the magnetization of SPIONs. In 2023, Li et al.[64] introduced a refined Jiles–Atherton (MJA) model for a more precise description of the dynamic magnetization of SPIONs. The MJA model is computed as shown in Equation 4 in Table 3. M_an is the Langevin function. M_MJA=M_irr is associated with the displacement of the domain wall. The effectiveness of image deblurring improves progressively from the X-space model to the Debye model, and further to the MJA model.

In conclusion, the primary benefit of the X-space reconstruction method is its speed, allowing for real-time imaging. However, due to the relaxation effects of particles, the reconstructed images suffer from blurring, and more accurate models are needed to describe the magnetization process of SPIONs.

SM-based reconstruction

SM-based image reconstruction is a technique that involves using a matrix to represent the mathematical relationship between the acquired signal and the desired image. The relationship between the particle concentration c(r) at the position r\in \Omega and the measured signal u(t) is shown in Equation 6 in Table 3. The signal u(t) and kernel s(r, t) can be expressed as Fourier series, as shown in Equation 6 in Table 3, which has the matrix form Sc=u.

Methods for acquiring the SM

The SM contains information about the characteristics of the imaging system and the physical processes involved.[58] The four methods for obtaining the SM are as follows:

The measurement-based method is considered the gold standard for obtaining an SM due to its successful application in numerous studies.[42] This method involves using a cubic delta phantom to traverse the entire FOV, and acquiring signals at different positions to construct the SM, as shown in the Figure 4(A). The primary drawback of this method is its long acquisition time. The small voxel size leads to low SNR, which can be enhanced by averaging across multiple cycles, thereby extending the calibration time. A significant benefit is that it eliminates the need for models of scanner hardware or particle behavior, as the SM addresses all system imperfections.

In 2010, Knopp et al.[65] proposed a model-based method, which uses an appropriate signal chain model to simulate the entire MPI system. This process entails modeling the magnetic fields, receiving coil sensitivity profiles, particle dynamic responses to excitation signals, and the receive chain transfer function for each channel.[42] The method was evaluated using 2D measurement data along a Lissajous trajectory, showing it can reconstruct images with quality similar to those obtained from calibration-derived SMs. However, approximating the particle’s dynamic response with a simple Langevin model still differs from the real scenario, so more accurate particle models are needed to develop a more precise SM.

Halkola et al.[66] proposed to obtain the hybrid SM using both simulation and calibration methods. The fundamental concept is depicted in Figure 4(B). This method introduces an offset field corresponding to the selection field, and delta samples are measured by switching the position of the offset field. Therefore, this method does not require moving the sample for each measurement, as in calibration-based methods. Furthermore, the improved SM allows for a better SNR by enabling the independent selection of the delta sample size from the voxel size.

In practice, SM is usually large and sparse due to the numerous voxels and detectors used in MPI imaging.[59] Knopp et al.[67] emphasized leveraging SM sparsity through various basis transformations for CS reconstruction, significantly reducing calibration scans. Each row of the SM is individually addressed to solve an inverse problem for reconstructing missing data. Weber et al.[68] proposed leveraging the spatial symmetries of the SM, demonstrating that for 2D imaging, acquiring just one quarter of the matrix is sufficient. The other three-quarters can be reconstructed by mirroring the measured portion and applying the corresponding symmetry factors. Additionally, Weber and Knopp[69] combined these methods, reducing the number of calibration scans needed for an accurate MPI SM, as shown in Figure 4(C). Methods leveraging low-rank tensor representations[70] and coded calibration[71] scenes have been developed, further enhancing efficiency. These algorithms not only decrease calibration time but also ensure high-quality image reconstruction.

Overall, different SM acquisition strategies involve trade-offs among reconstruction accuracy, calibration time, and implementation complexity, as shown in Table 4. Measurement-based methods generally provide the highest reconstruction accuracy because they capture the complete system response without requiring prior modeling of scanner hardware or particle dynamics, but they suffer from long calibration time and low SNR for small voxel sizes. Model-based methods significantly reduce calibration time and improve flexibility; however, their performance depends strongly on the accuracy of physical modeling, particularly particle dynamic responses. Hybrid methods attempt to balance these two approaches by combining partial measurements with simulations, thereby reducing acquisition time while maintaining relatively high reconstruction accuracy. In addition, symmetry-based, compressed sensing-based, and low-rank tensor-based calibration acceleration strategies further reduce calibration effort and storage requirements, but they typically introduce additional algorithmic complexity and may rely on assumptions such as matrix sparsity or spatial symmetry. Therefore, the choice of SM acquisition strategy should be determined according to the specific requirements of imaging accuracy, calibration efficiency, and system implementation constraints.

Methods for SM-based reconstruction

The SM-based reconstruction methods rely on the linear relationship between voltage signal and MNPs’ concentration distribution to complete the reconstruction task, and this linear relationship is represented by the SM, as shown in Equation 6 in Table 3. Both the MPI signal and the SM can be directly acquired on the MPI device, and hence, solving this linear inverse problem would allow for the reconstruction with the SM and the signal acquired on the same device. However, the SM-based reconstruction problem is typically ill-conditioned[42]. This mainly arises from the strong correlation between neighbouring columns of the SM, since the effective excitation field and the particle magnetization response vary smoothly within the field-free region. As a result, the MPI forward operator behaves similarly to a spatial convolution with a smoothing kernel, which attenuates high-frequency spatial components of the particle distribution. Consequently, the SM often has a large condition number, making the reconstruction process sensitive to measurement noise, where even minor perturbations in the signal can lead to noticeable deviations in the reconstructed image. In practice, this instability may manifest as amplified background noise, reduced spatial resolution, and reconstruction artifacts, especially under low SNR conditions or incomplete frequency selection. Therefore, for such ill-posed inverse problems[72,73], mathematical regularization terms are commonly introduced to stabilize the inversion process and improve reconstruction robustness. These regularization strategies typically incorporate prior assumptions about spatial characteristics of the particle distribution, such as sparsity and smoothness. Depending on the choice of regularization terms and optimization strategies, a variety of reconstruction algorithms have been developed.

The algebraic reconstruction technique (ART) is the predominant method employed in SM-based MPI reconstruction. The ART algorithm typically uses Tikhonov regularization to constrain the reconstruction image and employs the Kaczmarz algorithm for completing the reconstruction task.[43] The Tikhonov regularization term aligns with the smooth distribution characteristic of particle distribution, and the Tikhonov/Kaczmarz algorithm can complete MPI reconstruction in just a few iterations, achieving excellent performance in many studies.[42,43,46] However, while the method can suppress noise, its noise suppression capability is limited.[45] Additionally, due to the Tikhonov regularization term aligning with the smooth characteristic, the sample images reconstructed by the Tikhonov/Kaczmarz algorithm tend to reconstruct blurring edges, which affects the image quality.[45]

To enhance the noise suppression capability of the reconstruction algorithm, Storath et al.[45] introduced the non-negative fused lasso model (NF-Lasso) and applied it to MPI reconstruction. NF-Lasso uses L1 norm and total variation (TV) terms as regularization constraints. The L1 norm corresponds to the sparsity prior information, allowing the reconstruction algorithm to obtain sparse solutions. Hence it can easily suppress values to zero. TV regularization corresponds to the prior information that pixel values change significantly at edges and change little in smooth regions. Hence, it can control the difference in neighboring pixel values to suppress noise interference. The NF-Lasso algorithm, incorporating L1 and TV terms, demonstrates superior noise suppression and edge preservation compared to the ART method. However, the NF-Lasso algorithm uses gradient descent for reconstruction, which limits its efficiency.

The alternating direction method of multipliers (ADMM) is widely used for constrained optimization problems[74] and is applicable to MPI image reconstruction. ADMM-based MPI reconstruction algorithms can incorporate L1, TV, or a combination of both as regularization terms[75–77], and ADMM (L1 + TV) uses regularization terms similar to those of the NF-Lasso algorithm. Compared to the ART algorithm, the ADMM (L1 + TV) algorithm has stronger noise suppression capability, and compared to the NF-Lasso algorithm, it has higher reconstruction efficiency. However, both the ADMM (L1 + TV) and NF-Lasso algorithms are subject to the limitations of the TV and L1 regularization terms. The TV regularization term can cause staircase effect[11,77], and the reconstructed MPI image may be divided into several constant regions, which severely affects the visual quality of the reconstruction. The L1 regularization term can also cause discontinuities in the reconstructed image of continuous samples and potentially affect MPI quantitative performance. In 2024, Zhu et al.[78] proposed a novel non-convex regularization method, including the minimax concave (MC) and TV regularization term. MC penalty is used to provide nearly unbiased sparse constraints, and this method can also complete reconstruction with the ADMM algorithm for solving. The findings suggest that this method more effectively maintains the MPI quantitative characteristics than the L1 regularization term.

Although regularization-based reconstruction methods significantly improve the stability of SM-based MPI reconstruction, each regularization strategy introduces specific trade-offs that may affect reconstruction accuracy and quantitative reliability. For example, Tikhonov regularization effectively suppresses noise but often leads to oversmoothing and blurred structural boundaries, which may reduce spatial resolution. TV regularization enhances edge preservation; however, it may introduce staircase artifacts that divide smooth regions into piecewise constant areas. Similarly, L1-based sparsity constraints can improve noise robustness but may introduce discontinuities in regions with smoothly varying tracer distributions and potentially affect quantitative accuracy. Nevertheless, selecting appropriate regularization strategies requires balancing noise suppression capability, structural preservation, and quantitative fidelity according to specific imaging requirements.

In conclusion, regularization-based reconstruction techniques improve the stability of SM-based MPI reconstruction by incorporating prior assumptions such as smoothness and sparsity; however, these priors may also introduce reconstruction biases, including oversmoothing effects, staircase artifacts, or reduced quantitative fidelity. In summary, designing effective regularization strategies remains an important research direction for achieving high-quality, quantitatively reliable MPI reconstruction.

Application of Deep Learning in MPI

MPI signal processing

MPI signals are affected by different types of noise, such as background noise, thermal noise, and random noise. These noises significantly affect the imaging quality of MPI. To suppress noise interference, background subtraction is typically used to remove background signals.[53] Additionally, noise interference can be reduced through methods like filtering frequency by SNR.[55,79] Deep learning networks can learn noise characteristics to effectively suppress interference.

The denoising of MPI signals typically involves processing in both the time and frequency domains. Peng et al.[80] presented MDD-Net, a multi-scale dual-domain network designed to filter nonlinear magnetization signals in MPI. MDD-Net first filters the signals in the frequency domain. Then considering that high noise levels may make it difficult to recover signal information at certain frequencies, the time-domain signal is filtered after inverse discrete Fourier transform (IDFT). The MDD-Net exploits both MPI frequency domain and time domain information through multi-scale modules and attention mechanisms. Experimental results show that this approach exhibits better noise suppression capability compared to wavelet filtering and other denoising methods.

Wei et al.[81] introduced BSS-TFnet, an attention-based network that leverages both time-domain and frequency-domain information to suppress background noise in the time-frequency spectrum. BSS-TFnet utilizes self-attention to capture both local and global features in time-frequency dimensions, enhancing MPI signal recovery while minimizing background noise. Experiments highlight BSS-TFnet’s effectiveness in mitigating harmonic interference and Gaussian noise.

Considering that supervised learning algorithms require high-quality labels, Peng et al.[82] also proposed a self-supervised learning algorithm. They utilized the symmetric structural characteristics of MPI response signals in the time domain within each cycle and designed a gradient loss function to facilitate network convergence. This method effectively suppresses noise without requiring labels for model training.

While these methods demonstrate promising performance in suppressing specific noise types, most existing approaches are either constrained to predefined noise models or limited to single-domain feature learning. In practical MPI systems, noise characteristics are often dynamic and scanner-dependent, and frequency-domain components exhibit intrinsic correlations related to frequency indices and coil channels. Therefore, Sun et al.[83] proposed a transformer-based multidimensional information fusion network (MIFTnet) to capture cross-frequency and cross-coil relationships for real-time signal denoising, as shown in Figure 5(A). By leveraging attention mechanisms to model global dependencies, such approaches improve noise suppression robustness and enhance subsequent image reconstruction quality.

Deep learning has been extensively researched in the field of signal processing, including speech signal denoising[84], seismic signal denoising[85], demonstrating strong capability in modeling complex noise distributions. However, existing deep learning approaches for MPI signal denoising mainly rely on generic time-domain or frequency-domain feature extraction strategies and often do not explicitly incorporate MPI-specific physical knowledge. In practical MPI systems, the measured signal is closely related to magnetic field dynamics, nanoparticle magnetization responses, coil sensitivity profiles, and the harmonic structure of excitation frequencies. Integrating such physics-informed priors into network design—for example through frequency-aware feature embeddings, coil-channel correlation modeling, trajectory-dependent signal representations, or SM-guided constraints—has the potential to improve both reconstruction performance and model interpretability. Therefore, incorporating MPI-specific physical mechanisms into deep learning architectures represents an important future research direction for achieving more robust and quantitatively reliable signal processing.

SM-based reconstruction

Deep learning has been widely applied in SM-based reconstruction, primarily in two aspects: one focuses on the acquisition of the SM, specifically SM super-resolution, and the other addresses the solution of the inverse problem as illustrated in Equation 6 in Table 3. These two aspects will be discussed in detail below.

SM super-resolution

As there is currently no accurate forward model, the SM is primarily obtained through measurements. Measured calibration is considered the most precise SM, and inceasing the number of measurement grid points can enhance the spatial resolution of MPI reconstruction images. However, it will cause much time cost for SM acquisition. Yet, with changes in the environment and updates to devices, the characteristics of MPI signals may undergo alterations. Hence, updates of the SM are required, and it is crucial to reduce the acquisition time of the SM. Another benefit of reducing the SM acquisition time is to mitigate thermal noise interference. Increasing SM acquisition time can impact data quality due to noise level variations. Therefore, to reduce the acquisition time, some algorithms have been proposed to recover high-resolution SM.[69,86–89]

Traditional methods can reduce acquisition costs through CS techniques[69,86], but the reduction is limited. Deep learning approaches can now recover complete SM from data down sampled by up to 64 times, offering significant cost reduction benefits. In 2020, Baltruschat et al.[90] introduced a deep learning approach for SM reconstruction to enhance resolution without significantly increasing time costs. This approach converted all the SM into RGB images and utilized convolutional networks to achieve RGB image super-resolution. The super-resolution RGB images were then converted back into complex data for MPI reconstruction tasks. Results demonstrated that this approach greatly reduced SM acquisition time, and the super-resolution SM could be used for phantom data reconstruction, outperforming CS.

In 2022, Güngör et al.[88] introduced the first transformer model for SM super-resolution (TranSMS) and proposed a novel data consistency submodule to enhance the quality of super-resolution by combining with the physical model of MPI signals. Both simulation and experimental data demonstrate that TranSMS significantly improves SM recovery and MPI reconstruction, achieving acceleration rates of up to 64-fold in 2D imaging.

To mitigate the data requirements, Shi et al.[89] proposed a progressive training network for MPI 3D SM super-resolution (ProTSM) in 2023. This method employs pseudo-labeling for unlabeled pre-training datasets to initially train the SM super-resolution model, thereby minimizing data needs. Subsequently, the model is fine-tuned with precisely labeled data. It uniquely incorporates coil channel and frequency index for SM calibration, integrating frequency information into the model training process, as shown in Figure 5(B).

In 2025, Zhang et al.[91] proposed a frequency structure consistency (FSC) learning framework for high-resolution SM recovery. To address the limitations of existing super-resolution methods in preserving high-frequency signal structures, the authors introduced a frequency-aware loss function that enforces structural consistency across frequency components. The framework further incorporates a real–imaginary–magnitude (RIM) embedding strategy and a Swin Transformer-based architecture to model complex signal relationships. This study underscores the importance of integrating frequency-domain structural constraints into learning-based SM super-resolution. Besides, Guo et al.[92] proposed a multi-slice knowledge-driven SM calibration method (MKD-SM) that exploits the similarity between adjacent slices to improve high-resolution SM recovery from downsampled measurements, further enhancing calibration accuracy while reducing acquisition effort.

These approaches leverage deep learning strategies to enhance SM resolution and subsequently improve MPI spatial resolution without significantly increasing acquisition costs. Additionally, Schrank et al.[93] utilized the implicit neural representations for SM super-resolution, which does not require fixed specific upsampling ratios. Yin et al.[94] applied deep image prior (DIP) to MPI SM super-resolution, eliminating the need for labeled data.

Although deep learning-based SM super-resolution methods significantly reduce calibration time and improve reconstruction efficiency, they may introduce potential inconsistencies with the physical forward model if the recovered SM deviates from the true system response. Such inconsistencies may lead to reconstruction artifacts, reduced quantitative accuracy, or limited generalization across different scanners and acquisition conditions. To address these challenges, several recent studies incorporate physics-informed constraints into the learning process, such as frequency-structure consistency enforcement, coil-channel correlation modeling, and data-consistency modules guided by the MPI signal formation mechanism. In addition, hybrid training strategies combining measured calibration data with simulated priors have been explored to improve reconstruction fidelity. These approaches help ensure that the super-resolved SM remains consistent with the underlying physical imaging model while maintaining the advantages of reduced acquisition cost. Nevertheless, designing learning-based SM recovery methods that simultaneously achieve high resolution, strong physical consistency, and reliable quantitative reconstruction remains an important research direction in MPI. Future research should further explore physics-guided SM recovery frameworks that explicitly incorporate MPI signal generation mechanisms into network design to improve both interpretability and reconstruction reliability.

Deep learning-based inverse problem solving

SM-based reconstruction methods visualize MPI signals by solving the inverse problem, thereby generating the reconstructed image. Currently, deep learning approaches have been applied, including model-based networks, end-to-end networks, and DIP methods.

Reconstruction methods using model-based networks have been widely applied in fields such as MRI[36] and show potential for improving MPI reconstruction image quality. The approaches embed a deep learning network into traditional reconstruction algorithms, replacing some operations to enhance accuracy and efficiency of the reconstruction algorithm. Currently, model-based methods can be used to reconstruct real MPI phantom data. In 2022, Askin et al.[77] proposed a plug-and-play MPI reconstruction algorithm (PP-MPI) that incorporates a dense residual connection network as an image denoising module, substituting the noise suppression component in the ADMM-based reconstruction algorithm. Results show that the quality of PP-MPI reconstructed images surpasses traditional reconstruction methods. The Kaczmarz-PnP method, akin to PP-MPI, utilizes a plug-and-play (PnP) denoiser to substitute the traditional handcrafted priors in Kaczmarz.[95] In 2023, Güngör et al.[96] proposed a deep equilibrium network (DEQ-MPI), and two networks were adopted to replace the regularization and data consistency terms respectively which were used in traditional reconstruction algorithms. These networks were then embedded into the ADMM-based reconstruction algorithm. DEQ-MPI demonstrated outstanding reconstruction performance in simulation and phantom experiments. And DEQ-MPI was explored to 3D MPI reconstruction in 2024.[97] The above reconstruction algorithms use deep learning models to replace some submodules of traditional reconstruction algorithms, reducing reliance on regularization terms and parameter selection. However, same as the traditional reconstruction algorithms, model-based reconstruction algorithms are iterative and require empirically determined iteration counts, which limits the reconstruction efficiency.

In addition to model-based networks, the DIP algorithm is also an iterative deep learning method. In 2020, Dittmer et al.[98] applied DIP to the field of MPI reconstruction. The DIP method employs an untrained neural network that takes random noise as input to produce the reconstructed output. The reconstructed result is then multiplied by the SM, and the error between the product and the actual acquired signal values is used as the loss to drive iterative training of the model. The quality of DIP reconstruction results is initially improving and then deteriorating. Hence, to achieve the highest possible reconstruction quality, it requires manual early stopping before the model converges. This method is an iterative reconstruction approach that does not require pre-training the model or labeled data. However, each trained model can only be used to reconstruct a single MPI image, and each reconstruction requires retraining the model, which results in extremely low efficiency. The termination condition requires manual control, and reconstruction quality is highly subjective and unstable, often leading to poor-quality reconstructed images. In 2023, Huang et al.[99] incorporated L1 norm and total variation terms into the loss function of DIP, achieving better reconstruction performance than Tikhonov regularized Kaczmarz reconstruction method in simulation experiments.

A more direct deep learning reconstruction approach involves using end-to-end models to directly learn the transformation from MPI signals to MNP distribution images. In 2017, Chae et al.[100] employed an end-to-end network for reconstructing 1D simulated MPI data. The results showed that a single-layer neural network could reconstruct simulated MPI images, and using multiple layers further improved the quality of reconstructed MPI simulated data. However, due to the complexity noise of real signals, this approach could not achieve reconstruction of real MPI data. In 2022, von Gladiss et al.[101] simulated MPI magnetic field distributions using a customized magnetic particle spectrometer device and obtained two simulated SMs. One of the SMs was used to construct training datasets for a neural network and then the other one was used to validate the algorithm’s feasibility. Experimental results showed that compared to traditional MPI reconstruction methods, the algorithm resulted in less noise in the reconstructed images. Peng et al.[102] achieved reconstruction of real phantom data by making changes of the data, model, and training data, and the results demonstrated better reconstruction performance than other reconstruction algorithms. However, this method still requires retraining the model along with the changed SM.

Image post-processing

Deep learning can effectively mitigate image blur and noise in MPI reconstruction images. The selection field gradient affects the spatial resolution of MPI images, and higher gradients result in clearer images and better spatial resolution but result in low SNR. In 2022, Shang et al.[103] introduced a fusion dual-sampling convolutional neural network (FDS-MPI) for post-processing reconstructed images, utilizing low-gradient simulated MPI images as inputs and high-gradient simulated MPI images as labels, as shown in Figure 5(C). Driven by the data, FDS-MPI can effectively remove image blur and enhance MPI images’ spatial resolution. To reduce the need for paired data, Zhang et al.[104] introduced contrastive learning to the MPI image post-processing field. This approach does not require labeled paired blurry images, and instead, it only needs unmatched clear images to train the model, achieving MPI image deblurring. Unidirectional Cartesian trajectories cause anisotropic resolution, and Shang et al.[105] introduced the anisotropic edge-preserving network (AEP-net), utilizing asymmetric convolution to address anisotropy and an uncertainty region module to restore edge information.

To reduce noise in MPI images, Sun et al.[106] utilized a simulation method based on the SM to obtain training data and denoised the MPI images using RED-CNN. Wang et al.[107] presented the content-noise feature fusion neural network (CNFFNet). These methods utilized simulated MPI images and noise data from empty scans to create a dataset. Through learning the magnetic particle distribution and noise characteristics in MPI images, CNFFNet effectively achieves denoising. Tsanda et al.[95] introduced a deep learning-driven post-processing method to eliminate the necessity for manual parameter optimization. This approach uses a neural network to combine images reconstructed with different parameters into a single high-quality image. Moreover, obtaining multiple projection views for 3D MPI imaging is time-intensive, and using sparse views for reconstruction compromises image quality. To address this issue, Wu et al.[108] proposed a sparse view post-processing network called PGnet, which mitigates data scarcity issues in projection MPI reconstruction, and prevents stripe artifacts in the reconstructed results. In 2026, Peng et al.[109] proposed FCS-edNET which operates directly on post-reconstruction images and integrates convolutional layers with multi-head self-attention to jointly model local details and global contextual structures. A multi-scale denoising prior and a joint spatial–frequency loss were introduced to improve structural fidelity and robustness. This study illustrates the effectiveness of post-reconstruction deep learning strategies for improving MPI image quality without modifying hardware or reconstruction frameworks. Recently, Zhang et al.[110] proposed a Mamba-based framework (MPI-Mamba) for anisotropic image calibration and deblurring in MPI reconstruction. By introducing latent feature fusion and diffusion-based feature extraction, the method improves spatial resolution consistency and boundary restoration performance, demonstrating superior results on both simulated and real MPI datasets. To improve image quality under low tracer concentration conditions, Liu et al.[111] proposed a lightweight transformer-based denoising method that enhances feature extraction while reducing model complexity through a residual-local transformer structure, enabling reliable reconstruction at low iron concentrations and highlighting the potential of transformer-based approaches for low-SNR MPI imaging scenarios.

The application of deep learning in MPI is rapidly advancing, with continuous research dedicated to creating deep learning architectures and training methodologies for MPI. Deep learning models face several challenges. A significant challenge is the requirement for extensive, diverse datasets, which are scarce in the MPI domain due to high acquisition costs. Additionally, deep learning models may struggle to generalize across different MPI imaging systems, tracer types, and may be sensitive to variations in image quality or acquisition protocols. Finally, training these models typically requires significant computational resources.

MPI resources

Deep learning-based MPI reconstruction methods typically require diverse datasets for model training, validation, and benchmarking. Therefore, publicly available experimental datasets and simulation platforms play an essential role in supporting the development and evaluation of learning-based algorithms. In this section, we summarize representative MPI data resources and discuss their relevance to deep learning-based signal processing, SM recovery, and image reconstruction tasks. We have summarized the currently available data resources for MPI, including real data collected using devices and various platform resources for generating simulated data, as outlined below.

OpenMPI dataset

The scarcity and high expense of MPI scanners restrict access for many research teams. In response, Knopp et al.[112] developed the OpenMPI dataset to offer global researchers’ free access to a preclinical MPI measurement data acquired using a Bruker scanner (Ettlingen). Measurements were conducted using four different phantoms across three different imaging sequences, together with dedicated calibration datasets for each sequence used to configure the SM. These datasets provide standardized benchmarks for evaluating deep learning-based methods, including SM super-resolution, signal denoising, and image reconstruction algorithms, enabling quantitative comparison across different reconstruction strategies.

Currently, many studies have conducted experiments on the three phantoms: shape phantom, resolution phantom, and concentration phantom, which are widely used for assessing reconstruction accuracy, spatial resolution recovery capability, and quantitative performance. As for the rotational phantom, it has currently been used in the study on motion artifact suppression for dynamic tracer distribution imaging.[113] These datasets also provide opportunities for developing and validating learning-based approaches for multi-patch reconstruction and dynamic imaging tasks.

Simulation datasets

Due to the scarcity of real MPI data, which is insufficient to support the large data requirements for training deep learning models, there is a need to develop simulation datasets.

Professor Jie Tian’s team at the Key Laboratory of Molecular Imaging, Chinese Academy of Sciences, has developed an MPI resource platform featuring resources for magnetic nanoparticle development, imaging reconstruction algorithms, MPI imaging equipment and applications, an MPI simulation platform, and various data resources. Among them, the software platform MPI reconstruction framework (MPIRF) developed by Shen et al.[114] has frequently been employed in studies utilizing the X-space reconstruction method for generating simulation data, supporting the development and evaluation of learning-based reconstruction and image post-processing methods.

Another public MPI website, the Open-Source MPI project (OS-MPI), is administered by the Magnetic Resonance Physics and Instrumentation Group’s MPI team from Harvard Medical School and MIT. It includes platforms for the design, construction, and analysis of MPI technologies. These platforms enable researchers to generate customizable training datasets under different imaging conditions, which is particularly valuable for developing physics-informed learning strategies and validating algorithm robustness across acquisition settings.

Overall, these public datasets and simulation platforms provide essential infrastructure for training, validating, and benchmarking deep learning-based MPI reconstruction algorithms, and they play an important role in improving the reproducibility and comparability of learning-based methods across different research groups.

Conclusion

This review provides a comprehensive overview of the MPI reconstruction pipeline, from signal acquisition to image formation, and summarizes recent advances in deep learning-assisted signal processing, SM recovery, SM-based and X-space-based reconstruction, and image post-processing. These developments demonstrate the strong potential of deep learning to improve noise suppression capability, reconstruction accuracy, and computational efficiency in MPI.

Despite these advances, several technical challenges remain that limit the broader application of MPI, particularly in terms of spatial resolution, FOV scalability, calibration efficiency, and the difficulty of achieving stable real-time reconstruction in practical imaging scenarios. Future research should further explore physics-informed deep learning strategies that explicitly incorporate magnetic field dynamics, nanoparticle magnetization responses, and coil sensitivity characteristics into model design, enabling improved reconstruction fidelity and interpretability while preserving consistency with the MPI forward model.

In addition, data scarcity remains a critical limitation for learning-based MPI reconstruction methods, especially for SM calibration and high-resolution reconstruction tasks. Developing self-supervised, weakly supervised, and physics-constrained learning frameworks may help reduce dependence on large-scale labeled datasets and improve robustness across different acquisition conditions. Improving model generalization across scanners, acquisition trajectories, and tracer types is another important research direction, which may be addressed through hybrid model-based and data-driven reconstruction strategies.

Furthermore, integrating deep learning with SM extrapolation and calibration acceleration techniques provides a promising solution for reducing calibration cost and enabling efficient multi-patch imaging with reduced boundary artifacts. The joint optimization of reconstruction algorithms with acquisition trajectory design, frequency selection strategies, and hardware-aware calibration schemes is also expected to further enhance reconstruction stability and imaging efficiency.

Finally, achieving reliable real-time MPI reconstruction remains a key challenge for clinical transformation. Combining lightweight neural network architectures with physics-guided reconstruction frameworks and efficient frequency-domain signal representations may provide a practical pathway toward real-time imaging. In parallel, recent progress in large-FOV MPI system development, such as primate-sized scanners[115] based on digital-scanned focus field technology, demonstrates the feasibility of extending MPI toward clinically relevant imaging scales. Together with advances in tracer engineering and multimodal MPI hybrid imaging systems such as MPI–MRI integration, these developments are expected to promote the evolution of MPI toward large-FOV imaging and clinical-scale molecular imaging applications in precision medicine.

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The Author(s) 2026. This article is published by Higher Education Press at journal.hep.com.cn.